It is exemption Code "G" for 2018.  If you are unsure how to do that, hopefully a Lacerte user will come along (I use ProSeries).

https://www.irs.gov/pub/irs-dft/i8965--dft.pdf#page=4

However, that applies to 2018 ONLY.   From your second pdf:  "This change would apply to 2018 only; thus consumers will only be able to claim the hardship exemptions under §155.605(d)(1) on a federal income tax return for tax year 2018."

For prior years, they need to apply for an ECN through the Marketplace, including the "documentary evidence".  I also suspect that in order to get it for 2016, that would need to be done within the next two weeks.

As for your question of "what would be deemed to be sufficient in order to prove that this lack of choice has precluded them from obtaining coverage under a QHP?", the taxpayer would explain why that ONE provider did not work for their situation (for example, showing that their current doctors were not 'in network' of that provider).